Why does one never hear about a scalar addition operation? It would
make sense to me to define the following:

[1 2] [3 4]
[3 4] + 2 = [5 6]

Well, you could define such an operation, but mathematically it isn't
necessary; the above operation is equivalent to:

[1 2] [1 1] [3 4]
[3 4] + 2[1 1] = [5 6]

Defining an extra addition operation doesn't have any place in the
axioms governing vector spaces, either, so it would just be
'something extra'. Assuming it's defined as my above example
illustrates, I can't think of any way to break 'scalar addition'
without violating the underlaying ring axioms, so it should be solid
mathematically.

Similarly, why is there no scalar exponentiation on vectors?

Do you mean something like the following:

[1 2] [1 2][1 2][1 2]
( [3 4] )^3 = [3 4][3 4][3 4]

If that's the case, this sort of operation has been around for quite
some times, and stems from the set of NxN matrices being a ring. If
you're talking about non-square matrices, than exponentiation would
be undefined, because the product: